Complex Flows and Soft Matter Group

Schooling Wings

The complex collective dynamics of fish schools and bird flocks have long fascinated physicists, biologists, and mathematicians. In addition to their biological relevance, they are living examples of active systems in which energy input by the individual constituents gives rise to organized collective phenomena. While there has been considerable experimental and theoretical progress in characterizing ``dry” active systems (e.g., shaken granular rods) and the collective behavior of biological systems at the microscale (e.g., bacterial suspensions), significantly less is known about the role of hydrodynamic interactions in mediating schooling and flocking behavior in collectives of larger animals. More generally, the influence of inertial fluid flows and the consequent long-lived hydrodynamic interactions on collective behavior remains poorly understood.

The goal of this research program is to construct and validate a modeling framework for the long-lived hydrodynamic interactions between self-propelled swimmers, modeled as rigid flapping wings in a fluid. Our models consist of both discrete-time dynamical systems (iterated maps) and systems of partial differential equations (PDEs). The theoretical predictions of these models are validated against laboratory experiments and observations whenever possible.

Recent Publications



P. J. Baddoo, N. J. Moore, A. U. Oza, and D. G. Crowdy, Generalization of waving-plate theory to multiple interacting swimmers, Communications on Pure and Applied Mathematics, 76, 3811-3851, (2023)





Abstract Early research in aerodynamics and biological propulsion was dramatically advanced by the analytical solutions of Theodorsen, von Kármán, Wu and others. While these classical solutions apply only to isolated swimmers, the flow interactions between multiple swimmers are relevant to many practical applications, including the schooling and flocking of animal collectives. In this work, we derive a class of solutions that describe the hydrodynamic interactions between an arbitrary number of swimmers in a two-dimensional inviscid fluid. Our approach is rooted in multiply-connected complex analysis and exploits several recent results. Specifically, the transcendental (Schottky–Klein) prime function serves as the basic building block to construct the appropriate conformal maps and leading-edge-suction functions, which allows us to solve the modified Schwarz problem that arises. As such, our solutions generalize classical thin aerofoil theory, specifically Wu's waving-plate analysis, to the case of multiple swimmers. For the case of a pair of interacting swimmers, we develop an efficient numerical implementation that allows rapid computations of the forces on each swimmer. We investigate flow-mediated equilibria and find excellent agreement between our new solutions and previously reported experimental results. Our solutions recover and unify disparate results in the literature, thereby opening the door for future studies into the interactions between multiple swimmers.

A. U. Oza, L. Ristroph, and M. J. Shelley, Lattices of Hydrodynamically Interacting Flapping Swimmers, Physical Review X, 9, 041024, (2019)





Fish schools and bird flocks exhibit complex collective dynamics whose self-organization principles are largely unknown. The influence of hydrodynamics on such collectives has been relatively unexplored theoretically, in part due to the difficulty in modeling the temporally long-lived hydrodynamic interactions between many dynamic bodies. We address this through a novel discrete-time dynamical system (iterated map) that describes the hydrodynamic interactions between flapping swimmers arranged in one- and two-dimensional lattice formations. Our 1D results exhibit good agreement with previously published experimental data, in particular predicting the bistability of schooling states and new instabilities that can be probed in experimental settings. For 2D lattices, we determine the formations for which swimmers optimally benefit from hydrodynamic interactions. We thus obtain the following hierarchy: while a side-by- side single-row ``phalanx" formation offers a small improvement over a solitary swimmer, 1D in-line and 2D rectangular lattice formations exhibit substantial improvements, with the 2D diamond lattice offering the largest hydrodynamic benefit. Generally, our self-consistent modeling framework may be broadly applicable to active systems in which the collective dynamics is primarily driven by a fluid-mediated memory.

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S . Ramananarivo, F. Fang, A. Oza, J. Zhang, and L. Ristroph, Flow interactions lead to orderly formations of flapping wings in forward flight, Physical Review Fluids, 1, 071201, (2016)





Classic models of fish schools and flying formations of birds are built on the hypothesis that the preferred locations of an individual are determined by the flow left by its upstream neighbor. Lighthill posited that arrangements may in fact emerge passively from hydro- or aerodynamic interactions, drawing an analogy to the formation of crystals by intermolecular forces. Here, we carry out physical experiments aimed at testing the Lighthill conjecture and find that self-propelled flapping wings spontaneously assume one of multiple arrangements due to flow interactions. Wings in a tandem pair select the same forward speed, which tends to be faster than a single wing, while maintaining a separation distance that is an integer multiple of the wavelength traced out by each body. When perturbed, these locomotors robustly return to the same arrangement, and direct hydrodynamic force measurements reveal springlike restoring forces that maintain group cohesion. We also use these data to construct an interaction potential, showing how the observed positions of the follower correspond to stable wells in an energy landscape. Flow visualization and vortex-based theoretical models reveal coherent interactions in which the follower surfs on the periodic wake left by the leader. These results indicate that, for the high-Reynolds-number flows characteristic of schools and flocks, collective locomotion at enhanced speed and in orderly formations can emerge from flow interactions alone. If true for larger groups, then the view of collectives as ordered states of matter may prove to be a useful analogy.

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